Generative Deep Learning for the Two-Dimensional Quantum Rotor Model

TL;DR

This work designs two models based on the foundational architecture of generative adversarial networks to investigate the ground-state properties and phase transition characteristics of the two-dimensional quantum rotor model (QRM) within a semi-supervised learning framework.

Abstract

The advancement of diverse generative deep learning models and their variants has furnished substantial insights for investigating quantum many-body problems. In this work, we design two models based on the foundational architecture of generative adversarial networks (GANs) to investigate the ground-state properties and phase transition characteristics of the two-dimensional quantum rotor model (QRM). Within a semi-supervised learning framework, we incorporate multiple layers of transposed convolutions in the generator, enabling the conditional GAN to more efficiently extract low-dimensional encoded information. Analysis of one-dimensional latent variables associated with ground-state samples for different system sizes allows us to pinpoint the location of the critical point. In addition, we introduce dynamically adaptive weighting factors related to the distributional characteristics into the loss function of the deep convolutional GAN, and utilize upsampling techniques to enlarge the generated sample sizes. Comparisons of the optimization processes for mean magnetization and potential energy density across different magnetization regimes of QRM demonstrate that our model can efficiently generate valid ground-state samples, significantly reducing computational time. Our results highlight the promising potential of generative deep learning in quantum phase transition research, especially in critical point identification and the auxiliary generation of simulation data for quantum many-body models.

Figures (7)

• Figure 1: Schematic diagram of the conditional generative adversarial network structure with extractable generator-encoding information. The lower bracket illustrates the general CGAN training workflow, while the upper bracket highlights the details of the transposed convolutional operations added to the generator. The input data are preprocessed and successively passed through convolutional layers for dimensionality reduction and transposed convolutional layers for upsampling, with corresponding adaptations made to the discriminator for compatibility. After training, both two-dimensional and one-dimensional encoded representations are extracted from the generator for analysis.
• Figure 2: Optimization process of the ground-state energy for the two-dimensional QRM with $L = 4$. For different coupling parameters $g$, the sample energies gradually converge and stabilize after sufficient optimization steps, indicating that the system reaches its ground state. The optimized parameters corresponding to the ground state are then stored and used to generate a large number of samples for constructing the training dataset.
• Figure 3: After CGAN training on a series of ground-state samples of the two-dimensional QRM at various $g$ values, the generator’s 2-D encoded representations for the test set were extracted. Different colored points represent the discriminator’s predicted classes. Based on these classifications, the generator has learned to extract statistics that distinguish between the quantum paramagnetic phase and the quantum diamagnetic phase.
• Figure 4: (a) One-dimensional encoded information extracted from the generator for the system size $L \times L = 4 \times 4$, obtained from the test set composed of ground-state samples at various coupling parameters $g$. The red dashed line represents the polynomial fitting curve, while the blue line in the inset shows the curvature of the fitted curve. By analyzing the hidden variable $LV$, we find that the minimum curvature extremum effectively characterizes the location of the critical point, consistent with the physical behavior of the QRM, where spin orientations undergo drastic changes near the critical region. (b) One-dimensional encoded information from the generator for $L \times L = 12 \times 12$. Unlike Fig. \ref{['f_4']}(a), the fitted curve exhibits opposite monotonicity, and the maximum curvature extremum corresponds to the point of the most rapid change in the fitted curve, allowing the identification of the critical coupling $g_c$.
• Figure 5: Determination of the critical point and linear regression for system sizes $L = 4, 6, 8, 10, 12$. Based on the finite-size data of the two‑dimensional QRM, the y-intercept of the red line obtained from the fit provides a reasonable extrapolation of the critical point to the thermodynamic (infinite-size) limit.